MIDDLE SCHOOL MATH WORD PROBLEMS - METHOD OF SOLVING, EXAMPLES, EXERCISE

Please study

Linear Equations in Two Variables before Middle School Math Word Problems

if you have not already done so.

There, we explained the method of solving the equations, with examples.
That knowledge is a prerequisite here.

Here, we apply that knowledge to solve word problems.

Word Problems involving Linear Equations in two variables

The steps involved in the method of solving a word problem
involving Linear Equations in two variables are given below.

Step 1 :

Read the problem carefully and note down
what are given and what are required.

Step 2 :

Select two letters say x and y to represent
the unknown quantities asked for.

Step 3 :

Represent the word statements of the problem
in the symbolic language step by step.

Step 4 :

Look for quantities which are equal as per
conditions given and form two equations.

Step 5 :

Solve the two linear equations in two variables
obtained in step 4.

Step 6 :

Check the result for making sure that your
answers satisfy the requirements of the problem.

Example 1 of Middle School Math Word Problems

Solve the following Problem from Middle School Math Word Problems.

5 years ago, A was thrice as old as B and 10 years later, A shall
be twice as old as B. What are the present ages of A and B ?

Solution to Example 1 of Middle School Math Word Problems :

Let x and y be the present ages of A and B respectively.
5 years ago their ages will be x - 5 for A and y - 5 for B.
5 years ago, A was thrice as old as B
⇒ x - 5 = 3(y - 5) = 3y - 15 ⇒ x - 3y = -15 + 5 = -10
⇒ -x + 3y = 10....................................(i)
10 years later their ages will be x + 10 for A and y + 10 for B.
10 years later, A shall be twice as old as B
⇒ x + 10 = 2(y + 10) = 2y + 20 ⇒ x - 2y = 20 - 10
⇒ x - 2y = 10....................................(ii)

Equations (i) and (ii) are the Linear Equations
in two variables formed by converting the given
word statements to the symbolic language.

Now we have to solve these simultaneous equations.
(i) + (ii) gives 3y - 2y = 10 + 10 ⇒ y = 20.
Using this in (ii), we get
x - 2(20) = 10 ⇒ x -40 = 10 ⇒ x = 10 + 40 = 50.
Thus A is 50 years old and B is 20 years old. Ans.

Check:
5 years ago, A's age = 50 - 5 = 45; B's age = 20 - 5 = 15
5 years ago, A's age = 45 = 3 x 15 = thrice B's age. (verified.)
10 years later, A's age = 50 + 10 = 60; B's age = 20 + 10 = 30
10 years later, A's age = 60 = 2 x 30 = twice B's age. (verified.)

Example 2 of Middle School Math Word Problems

Solve the following problem from Middle School Math Word Problems.

The difference of the digits of a two-digit number is 3. If 4 times
the number is equal to 7 times the number obtained by reversing
the digits, find the number.

Solution to Example 2 of Middle School Math Word Problems

By data,
4 times the number = 7 times the number with the digits reversed.
⇒ the number with the digits reversed < the original number.
⇒ In the original number, ten's digit > unit's digit.
Let x be the ten's digit and y be the unit's digit in the original number.
Then x > y. Their difference is 3 means x - y = 3........................(i)
The original number = y + 10x
The number with the digits reversed = x + 10y
4 times the number = 7 times the number with the digits reversed.
⇒ 4(y + 10x) = 7(x + 10y) ⇒ 4y + 40x = 7x + 70y
⇒ 40x - 7x = 70y - 4y ⇒ 33x = 66y
⇒ x = 2y.......................(ii)

Equations (i) and (ii) are the Linear Equations
in two variables formed by converting the given
word statements to the symbolic language.

Now we have to solve these simultaneous equations.
Using (ii) in (i), we get
2y - y = 3 ⇒ y = 3
Using this in (ii), we get
x = 2y = 2(3) = 6.
∴ The required number = 63. Ans.

Check:
The difference of the two digits = 6 - 3 = 3. (verified.)
4 times the number = 4(63) = 252.
7 times the number with the digits reversed = 7(36) = 252
= 4 times the number (verified.)

Exercise on Middle School Math Word Problems

Solve the following Problems from Middle School Math Word Problems.

1. A two-digit number is seven times the sum of its digits.
   The number formed by reversing the digits is 18 less than the
   original number. Find the original number.
   
2. The result of dividing a number of two digits by the number
   with digits reversed is 5⁄6. If the difference of digits is 1,
   find the number.
   
3. If the numerator of a fraction is increased by 2 and its denominator
   decreased by 1, then it becomes 2⁄3. If the numerator is
   increased by 1 and denominator increased by 2,
   then it becomes 1⁄3. Find the fraction.
   

Answers to Exercise on Middle School Math Word Problems

The Answers to the Problems from
Middle School Math Word Problems given
in the exercise above, are given below.

1. 42
2. 45
3. 2⁄7